Egg cartons line the bench, each cell a seedling pot. A jar of dried beans stands in for seedlings.
"Lab day," Comet says. "Count a handful, make rows, record the leftover. Then change the row size."
Wren tips out a pile and counts. "Seventy-three. Rows of five." He lays them out. "14 rows, 3 left."
"Now rows of twelve," Comet says. They rebuild. "Eight rows, two left. What do you notice?"
"Same pile, different remainder," Wren says. "The remainder belongs to the pair, pile and row."
Nova records every count. "Would you like a hint? Your tally rule p(x) is the pile. The divisor is the row."
"So p(a) is the leftover when rows are x - a," Comet says. "Let us test it with the cartons."
A grown-up carries the big bean jar to the bench for them.
Every count is the crew's own tally, made up for the Glass House. Yours will differ.
| Pile n | Row size k | Full rows | Left over | Check |
|---|---|---|---|---|
| 61 | 4 | 15 | 1 | 4 × 15 + 1 = 61 |
| 73 | 5 | 14 | 3 | 5 × 14 + 3 = 73 |
| 98 | 12 | 8 | 2 | 12 × 8 + 2 = 98 |
| 45 | 9 | 5 | 0 | 9 × 5 + 0 = 45 |
The last row has no leftover: 45 is 9 × 5 exactly. In polynomial words, the divisor is a factor.
A zero remainder is the special case. The whole pile fits in rows, and the whole polynomial splits into factors.
| Divisor | Quotient | Remainder | p(a) |
|---|---|---|---|
| x + 2 | x² + 4x + 3 | 1 | 1 |
| x - 1 | x² + 7x + 18 | 25 | 25 |
| x + 3 | x² + 3x + 2 | 1 | 1 |
| Statement | True or false? |
|---|---|
| 14 × 5 + 3 = 73 | ? |
| 98 - 8 × 12 = 2 | ? |
| The same pile always gives the same remainder, whatever the row size. | ? |
| (0 - 2) × (0 - 2) × (0 - 2) + 6 × 4 - 22 + 7 = 1 | ? |
| A remainder of 0 means the divisor is a factor. | ? |
Good counting. Tomorrow you prove the remainder theorem and use it to factor a cubic completely.